Database Reference
In-Depth Information
=
k
∩
l
2
◦
h
2
∩
f
k
◦
T
(l
2
◦
h
2
)
◦
f
◦
in
OP
=
in
f
OP
=
T
e
(l
1
◦
h
1
;
l
2
◦
h
2
)
T
e
(l
1
;
h
2
)
.
=
l
2
)
◦
(h
1
;
Thus,
T
e
((l
1
;
T
e
(l
1
;
T
e
(h
1
;
◦
(h
1
;
=
◦
h
2
)
.
Let us show this functorial composition property also when we have the complex
arrows. We have (from Lemma
11
):
(i)
in
l
2
)
h
2
))
l
2
)
in
OP
Tg
OP
.
◦
g
=
g
OP
Thus,
T
e
(l
1
;
T
e
(h
1
;
in
OP
in
OP
l
2
)
◦
h
2
)
=
k
◦
T(l
2
◦
g)
◦
in
g
OP
◦
g
◦
T(h
2
◦
f)
◦
in
f
OP
in
OP
Tg
OP
=
k
◦
T(l
2
◦
◦
◦
T(h
2
◦
◦
g)
f)
in
f
OP
from
(
i
)
T
l
2
◦
f
◦
in
OP
g
OP
=
k
◦
◦
◦
h
2
◦
g
in
f
OP
T
l
2
◦
g
g
◦
h
1
◦
in
OP
g
OP
=
k
◦
◦
◦
in
f
OP
(
since
h
2
◦
f
=
g
◦
h
1
)
in
OP
=
k
◦
T(l
2
◦
g
◦
h
1
)
◦
in
f
OP
(
from Lemma
11
)
k
◦
T
(l
2
◦
h
2
)
◦
f
◦
in
OP
=
in
f
OP
(
since
h
2
◦
f
=
g
◦
h
1
)
=
T
e
(l
1
◦
h
1
;
l
2
◦
h
2
)
=
T
e
(l
1
;
l
2
)
◦
(h
1
;
h
2
)
.
For any identity arrow
id
A
,
T
e
J(id
A
)
=
id
A
=
TA
A
. Thus, as required by Defi-
nition
5
, the natural isomorphism
ϕ
I
DB
is valid.
Let us show that
DB
is an
extended symmetric
category, i.e.,
τ
−
1
:
T
e
◦
•
=
:
τ
ψ
−→
S
nd
for vertical composition of natural transformations
τ
:
F
st
−→
T
e
and
F
st
−→
τ
−
1
:
T
e
S
nd
, so that for each object
J(f)
=
A,B,f
in
DB
↓
DB
, the follow-
ing diagram commutes (we recall that
ψ
J(f)
=
ψ(J(f))
=
f
),
where
f
=
T
e
J(f)
=
f
f
ji
,
f
f
ji
∈
